Explore BrainMass
Share

# Algebra

### 1. Let N1(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic residue and the second is a quadratic nonresidue modulo p. Prove that N1(p) = (1/4) (p - ( - 1)^((p - 1)/2)) 2. Let N2(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic nonresidue and the second is a quadratic residue modulo p. Prove that N2(p) = (1/4) (p - 2 + ( - 1)^((p - 1)/2)) 3. Let N3(p) denote the number of pairs of integers in [1, p - 1] in which the first is a quadratic nonresidue and the second is a quadratic nonresidue modulo p. Prove that N3(p) = (1/4) (p - 2 + ( - 1)^((p - 1)/2)) 4. Use the results of theorem 10-4 and corollary 10-1 to construct solutions of x^2 + y^2 =29. 5. Prove ( without assuming corollary 10-1) that, if p is a prime &#8801; 1 (mod 4), then there exists positive integers m, x, and y such that x^2 + y^2 =mp, with p &#9532; x, p &#9532; y, 0 < m < p [ Hint: use the proof of Theorem 11-2]. Theorem 10-4: If p is an odd prime, then &#957;(p) = (1/8)p + Ep where | Ep| < (1/4)(p)^(1/2) +2. Corollary 10-1: Every prime p &#8801; 1(mod 4) is representable as a sum of two squares. Theorem 11-2: For each prime p there exist integers A,B and C, not all zero, such that A^2 + B^2 + C^2 &#8801; 0 (mod p).

Theory of Numbers The Distribution of Quadratic Residues Sums of Squares Sums of Four S

### Solving an Equation by Substitution

Please solve: v=ab/(c+d) where v = velocity where a = 6.67*10^-11 N m^2 kg^-2 where b = 5.97*10^24kg where c = 6.37*10^3km where d = 385km could you show all workings and all terms in the equation should be in SI units

### Solving an Equation by Substituting Values

If air resistance is neglected, the number of feet "s" that an object falls during "t" seconds is given by the equation s = 16t^2. How long would it take for an engine of an airplane to reach the earth if it fell off the airplane at 32,000 feet?

### Shapley-Shubik Distribution, Weighted Voting Systems and Combina

1) Find the Shapley- Shubik distribution (in percent) for the system (7: 5,3,2,1,). 2) In a cafeteria every meal has a 5 choice main course, a 3 choice salad, and a 2 choice soup, and a 4 choice dessert. If some one ate a different complete meal every day, how many days could he eat before he would start to repeat a meal?

### Solve an Equation Word Problem

The formula 1211.1L = CA squared S is used to determine the approach speed for landing an aircraft, where L is the gross weight of the aircraft in pounds, C is the coefficient of lift, S is the surface area of the wings in square feet (ft squared), and A is the approach speed in feet per second. I need help finding A for the Pi

### Rectangles : Solving for the Golden Ratio

If the ancient Greeks thought that the most pleasing shape for a rectangle was one for which the ratio of the length to the width was 8 to 5, and the golden ratio. If the length of a rectangle painting is 2 feet longer than its width, then for what dimensions would the length and the width have the golden ratio?

### Solve the Equation and Writing a Word Problem as an Equation

1. solve 2(3y+4)-5(y-2)=3y-6(y-3) 2. Four more than twice the difference of three times a number and 8 is less than twice the number.

### Solve the equation

Z/z-2 - 1/z+5 = 7/z squared + 3z - 10

### Ratio

The student-teacher ratio for Stratford High was reported to be 27.5 to 1. If there are 42 teachers, how many students are in the school?

### Complex Variable Class - Undergraduate 500 Level

Compute c&#8747; (e^cosz)(logz)dz where c is the positive oriented circle with center z0=1 and radius 1/2. Please see attached.

### Two-Step Equations and Equations with Variables

The question is, can you help me solve problems like: 5 - 2n = 8n -(-4n) ?

### How do you solve an equation?

In math class we have been assigned to answer the following question: How do you solve an equation? Please provide an example. I know how to solve an equation if it is given to me, but I do not know how to describe how to solve any equation. I hope you can help.

### Quadratic Reciprocity Law

Prove, using the quadratic reciprocity law that (3/p) = 1 if p== 1 or 11 (mod 12) (3/p) = -1 if p==5 or 7 (mod12) for each odd prime p.

### Writing an Equation from a Word Problem : 'Separately and Both Together' Problem

With only the small valve open, all of the liquid can be drained from a large vat in 4 hours. With only the large valve open, all of the liquid can be drained in 2 hours. How long would it take to drain the vat with both valves open?

### Minimum Cost Problems for Company C

Minimum cost problem: It cost a company C dollars per hour to operate its base base division. an analyst had determined that C is related to the number of base balls produced per hour, x, by the eqauation C = 0.009x^2 - 1.8^2 + 100. What number of balls per hour should be produced to minimize the cost per hour of manufacturing

### Capacitor, Source Voltage and Maximum Charge

If a capacitor with a capacitance of 0.000005 farads is wired in a circuit with a total resistance of 5000 ohms (the standard unit of electrical resistance), how long must the capacitor be wired to a source voltage to reach 50% of its maximum charge? (Hint: Begin by expressing q as 0.5Q)

### Word Problem : Application of Sums of Odd and Even Numbers

As Jack finished a section of the novel he was reading, his wife posed a curious question: "Dear, suppose you took the page numbers of the section you just read and added them together. What would you get? Jack summed them rather hastily and said, "Well, I may have to double-check my arithmetic. It's either 412 or 512, I'm not

### Word Problem : Medical Insurance Co-Payments

On the medical insurance policy (such as BCBS), the policy holder may need to make co-payments for percriptions drugs, office visits, and proceduares until total of all co-payments reaches a specific amount. Suppose on the Gattelaro's medical policy that the co-payment for the percription drug is 50% of the cost; the co-payment

### Word Problem : Speed and Distance

Ryan starts jogging home at the rate of 6km/hr. After reaching Harbor Front, he begins jogging back home. Because of fatigue, his speed for the return trip decreases to 4.5km/h and takes 1 hour longer. Determine the total distance that Ryan jogged.

### Formula to determine variable of gross profit %

Attached is a spreadsheet with two accounts. Each account has the same retail and invoice. One school has a small enrollment and the other has a larger enrollments. There is a handling fee of \$2 per enrollment. There are only two variables that can be changed: brochure cost & incentives. Also, the larger the enrollment, the larg

### Line Equations :Find Equation Given Point and Perpendicular Line and Find Point

1: Write the standard form of the equation of the line passing through the point (1,-2) and perpendicular to the line y=-3/2x-2 2: The line through A(-3,-4) and B(-2,Y) is perpendicular to a line with a slope -1/7. Find y

### Bounded Real Sequences and Metrics

Let X be the set of all bounded real sequences ... (SEE ATTACHMENT FOR COMPLETE QUESTION) ... For points x = {x1,,x2,x3...} and y = {y1,y2,y3...} in X define: d(x,y) := SUP |xk-yk|. (k less than or equal to 1) Prove that d is a metric (remember to explain why d(x,y) is finite!) (Please explain in your own words

### Word Problem : Writing a Defining Equation and Solving the Equation

To stay competitive with the fast food restaurant across the street, the Pico de Gallo Restaurante offers a lunch special. Each lunch plate has an .80 charge for beans, rice and guacamole, and each taco oredered costs .75. Senora Taquita, across the street, only charges .35 for the beans, rice and guacamole, but charges .90 fo

### Linear Programming / Sensitivity Analysis / Optimization: Please solve in Excel using Solver add-in and also do Sensitivity Report to solve and support answers. The EnergyBoat Company

The Background The EnergyBoat, Inc. produces and distributes high quality motor boat with highly seasonal demand. To handle this seasonality, the company has several options for its supply chain aggregated planning purposes: hiring an additional workers during peak seasons, outsourcing part of the demand to a third party manu

### Counter Images of a Function

Please see the attached file for the fully formatted problems.

See attachment.

### Inner Product Space Convergence

Problem: If is a bounded sequence in an inner product space, and is a sequence converging to zero, prove that (see attachment). Note, here < , > is the inner product notation. Hint: Use triangle inequality.

### Word Problem

If Bob had \$150 more saved he would have five times as much saved as he currently has saved. How much does Bob currently have saved?

### Equation of sphere

Find an equation of the largest sphere with center (10, 1 , 3) that is contained completely in the first octant. = 0 Note that you must move everything to the left hand side of the equation that we desire the coefficients of the quadratic terms to be 1.

### Findinf the Equation of a Sphere Given the Center

Find an equation of the largest sphere with center (10, 1 , 3) that is contained completely in the first octant. Note that you must move everything to the left hand side of the equation that we desire the coefficients of the quadratic terms to be 1.